The Cantor's set theory is a Trojan Horse of the mathematics-XX: onthe one hand, it is a natural, visual, universal language to describemathematical objects, their properties and relations, originating fromthe famous Euler's "logical circles", and just therefore this languagewas accepted by all mathematicians with a natural enthusiasm. However,on the other hand, together with the language, Cantor's transfiniteconceptions and constructions (like the actualization of all infinitesets, a distinguishing of infinite sets by the number of theirelements (i.e., their cardinalities), the hierarchy of ordinal andcardinal transfinite numbers, continuum hypothesis, etc.) went intothe mathematics-XX. Just the Cantor's actualization of infinite setsgenerated a lot of set-theoretical paradoxes and, ultimately, theThird Great Crisis in foundations of mathematics in the beginning ofthe XX c. The theme itself of the present conference shows that theproblem of the actual infinity is not closed and the Third GreatCrisis in foundations of mathematics goes on hitherto.